Surface magneto-plasmons in magnetic multilayers - Walther ...

Section 2.2
Dispersion relation 7
where i represents the layer, the metal (1) and the dielectric (2). In the situation
discussed here Ey can again without any loss of generality treated as zero.
Furthermore, the magnetic field must be perpendicular to the electric field, the mag-
netic field has only a component in y-direction
H (i) (r, t) =
⎛
⎜
⎝
0
H (i)
y
0
⎞
⎟
⎠ e i(k(i) x x − ωt) ∓ik
e (i)
z z
(2.4)
Here, the − sign in the exponent of the evanescent part (e ∓ik(i)
z z )is valid for the metal
(z < 0) and the + sign for the dielectric (z > 0). It has to be denoted that the
surface plasmons propagate in positive and negative x-directions. However, since
both directions are symmetrical only the positive direction is considered, i.e. kx > 0.
Moreover, as the interface lies at z = 0 it is assumed that kz > 0 for both directions
of z.
To get the dispersion relation for surface plasmons the fact is used that both the
electric (Eq. (2.3)) and magnetic field (Eq. (2.4)) have to fulfil the Maxwell equations
in vacuum [24].
∇ × E (i)
= − ∂B(i)
∂t
∇ × H (i) = ∂D(i)
∂t
= −µ0µ (i) ∂H(i)
∂t
(i) ∂E(i)
= ε0ε
∂t
(2.5)
(2.6)
ε0ε (i) ∇E (i) = ∇D (i) = 0 (2.7)
µ0µ (i) ∇H (i) = ∇B (i) = 0 (2.8)
with ε0 and µ0 being the dielectric and magnetic permeability in vacuum and ε (i) and
µ (i) being the dielectric and magnetic permeability in a medium i. The solutions have
to satisfy the continuity conditions at the interface.
E (1)
x = E (2)
x
H (1)
x = H (2)
x
(2.9)
(2.10)
ε (1) E (1)
z = ε (2) E (2)
z . (2.11)